U,={ "EM ee (13.23) 
-a ifX,2d 
then the model X; can be expressed by a threshold model 
(1) 
Ife X,t+ Er, for X; <d 
Keele 170) XGCHE ee LORXGue= a ae 
where 
GD) 
{ 20 =a+ac (13.25) 
a’=a-Q Cc. 
This model gives a nonlinear process as shown by Eq. 13.22. It can be called a piece— 
wise linear approximation. As will be shown later in Section 13.5, Ozaki?! made it clear 
that this piece—wise linearization is a special case of his nonlinear threshold autoregres- 
sive model. 
13.4 EXPONENTIAL AUTOREGRESSIVE MODEL 
The nonlinear random vibration system of one degree of freedom is expressed as 
X(t) + 21{X(0)} - X@) + 2o{XO} XO = n(1) (13.26) 
where n(f) is a random noise excitation. 
When 
X@ = YO, (13.27) 
Eg. 13.26 becomes 
¥(t) =— go{X()} X()— 21 [X(D)} YOO). (13.28) 
From these two equations and the state space expression as explained in Section 12.3 
X(t) 0 1 X(t) 0 
Yi) |=|-g.xo} -six@}| | vo|* | no G32”) 
Or, in vector notation, 
V(t) = f{xeo} V(t) +n(0), (13.30) 
where 
V(t) = ah = a (13.31) 
So, finding the appropriate expression for g)(-), 22(-) in Eq. 13.28 is the same as finding 
the appropriate function f {x@} for Eq. 13.30. When these two functions g)(-), g2(-) are in 
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